an:01354804
Zbl 0936.34023
Andres, Jan; Gabor, Grzegorz; G??rniewicz, Lech
Boundary value problems on infinite intervals
EN
Trans. Am. Math. Soc. 351, No. 12, 4861-4903 (1999).
00061274
1999
j
34B40 34A60 34B15 34G25 47H04 54C60
asymptotic boundary value problems; differential equations and inclusions; topological methods; Fr??chet spaces
The authors develop two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals.
After historical remarks in Section 1, asymptotic boundary value problems as fixed point problems in Fr??chet spaces are considered in Section 2. The authors construct a generalized topological degree for set-valued mappings defined on subsets of Fr??chet spaces and a fixed point index on retracts of Fr??chet spaces. Using this they generalize and extend some existence statements due to the Florence group of mathematicians. For example the convexity restrictions in the Schauder linearization device can be avoided. Four examples complete Section 2.
In Section 3, the existence of bounded solutions to partially dissipative differential inclusions are proved on the positive half-line. Following the ideas of Andres, G??rniewicz and Lewicka, who have considered periodic problems, the authors use a sequential approach which consists in investigating the limit process for the family of boundary value problems on infinitely increasing compact intervals. Three examples are given, here.
In Section 4, the structure of solution sets for the Cauchy problem is investigated both for differential inclusions with u.s.c. and l.s.c. right-hand sides.
Sections 5 and 6 consist of some remarks on implicit differential equations on noncompact intervals and of open problems. A large list of references can be find at the end of the paper.
I.Rach??nkov?? (Olomouc)